# Course

## M554 - Ergodic Theory

Course No:
M554
Credit:
4
Approval:
2014
UG-Elective
Syllabus:
Measure preserving systems; examples: Hamiltonian dynamics and Liouvilles theorem, Bernoulli shifts, Markov shifts, Rotations of the circle, Rotations of the torus, Automorphisms of the Torus, Gauss transformations, Skew-product, Poincare Recurrence lemma: Induced transformation: Kakutani towers: Rokhlins lemma. Recurrence in Topological Dynamics, Birkhoffs Recurrence theorem, Ergodicity, Weak-mixing and strong-mixing and their characterizations, Ergodic Theorems of Birkhoff and Von Neumann. Consequences of the Ergodic theorem. Invariant measures on compact systems, Unique ergodicity and equidistribution. Weyls theorem, The Isomorphism problem; conjugacy, spectral equivalence, Transformations with discrete spectrum, Halmosvon Neumann theorem, Entropy. The Kolmogorov-Sinai theorem. Calculation of Entropy. The Shannon Mc-MillanBreiman Theorem, Flows. Birkhoffs ergodic Theorem and Wieners ergodic theorem for
flows. Flows built under a function.
Reference Books:

Peter Walters, “An introduction to ergodic theory”, Graduate Texts in Mathematics, 79. Springer-Verlag, 1982.Patrick Billingsley, “Ergodic theory and information”, Robert E. Krieger Publishing Co., 1978.M. G. Nadkarni, “Basic ergodic theory”, Texts and Readings in Mathematics, 6.Hindustan Book Agency, 1995.H. Furstenberg, “Recurrence in ergodic theory and combinatorial number theory”, Princeton University Press, 1981.K. Petersen, “Ergodic theory”, Cambridge Studies in Advanced Mathematics, 2.Cambridge University Press, 1989.